Defining Entropy (S) through the Clausius Inequality

Consider two heat engines, one a reversible
(Carnot) engine and the other an irreversible heat engine. For
purposes of developing the Clausius Inequality we assume that
both engines are sized to accept the same amount of heat QH from the thermal source. Thus since the irreversible
engine must be less efficient than the Carnot engine, it must
reject more heat QL,irrev to
the thermal sink than that rejected by the Carnot engine QL,rev , as shown:

Consider first the reversible (Carnot) heat
Engine. We saw in Chapter 5
that reversible heat transfer can only occur isothermally, thus
the cyclic integral of the heat transfer divided by the temperature
can be evaluated as follows:

Recall from Chapter
5 that whenever we considered the efficiency of a reversible
heat engine, we went into "meditation mode", replacing
the ratio of heat flows with the ratio of temperatures:

Notice from the above diagram showing the two
heat engines that for an irreversible engine having the same value
of heat transfer from the thermal source QH as the reversible engine, the heat transfer to the
thermal sink QL,irrev> QL,rev.
Let Qdiff = (QL,irrev- QL,rev), then the cyclic integral for an irrevesible heat
engine becomes:

Thus finally, for any reversible or irreversible
heat engine we obtain the Clausius Inequality:

Defining the property Entropy - S

All properties (such as pressure P, volume
V, etc) have a cyclic integral equal to zero.